Jesus instead gives a response clearly intended for the woman to hear: "I was sent only to the lost sheep of the House of Israel." This is not an answer to the disciples' request. Instead it is a rhetorical statement of something the disciples and the woman both know,* but it functions as a challenge to the woman: "Tell me why I should help you." Instead of sending her away, Jesus engages Himself in the interaction. She then is encouraged enough to come right up to him and kneel, switching from the beggar's standard plea to the simple words, "Lord, help me."
And here is what is truly astonishing. DeSilva tells us that a challenge of the sort Jesus offers was a common social interaction in ANE honor-shame cultures-- but only for men. He explains:
"[H]onor can be won and lost in what has been called the social game of challenge and riposte. It is this “game,” still observable in the modern Mediterranean, that has caused cultural anthropologists to label the culture as “agonistic,” from the Greek word for “contest”. The challenge-riposte is essentially an attempt to gain honor at someone else’s expense by publicly posing a challenge that cannot be answered. When a challenge has been posed, the challenged must make some sort of response (and no response is also considered a response). It falls to the bystanders to decide whether or not the challenged person successfully defended his (and, indeed, usually “his”) own honor. The Gospels are full of these exchanges, mainly posed by Pharisees, Sadducees or other religious officials at Jesus, whom they regarded as an upstart threatening to steal their place in the esteem of the people." p. 29, emphasis added.
The rest of the exchange between Jesus and the Syro-Phoenician woman is just this sort of challenge-riposte.